He kept the momentum going with a par save from the trees at nine before a birdie at 13.
Woods got up and down for birdie at 15, where his chip on left him two feet, and he reached 11-under with a seven-foot birdie at the par-three 16th.
"I just did everything," Woods said. "I drove it well and hit my irons well. I made some putts. I just let the round just kind of build."
Woods's effort thrilled the galleries at Augusta -- and got the attention of US President Donald Trump, who has played in the past with the superstar.
"Great playing by @TigerWoods (at the) @TheMasters," Trump tweeted. "Tomorrow will be a big and exciting day for golf and for sports ... Good luck to all!"
Finau, aiming to become the fifth straight player to make the Masters his first major title, opened with three straight birdies.
Another birdie at the sixth was followed by an eagle at eight, where his second shot left him a tap-in.
"I hit a really good shot, but I didn't think it was going to be, you know, a foot from the hole," Finau said.
Two more birdies coming in put Finau in a "dream" final group with Woods and Molinari -- after Augusta National officials announced the final round would be played in threesomes from both the first and 10th tees, starting early in a bid to beat severe weather predicted for the afternoon.
"As a kid I always wanted to compete against him," said 29-year-old Finau of Woods. "I've dreamed of playing in a final group with him in a major championship."
Molinari played with Woods in the final round at Carnoustie last year, when he lifted the Claret Jug.
"He obviously loves this place and he's playing great golf," Molinari said of Woods. "So I'm aware that it's not going to be easy.
"But it's not like I can only worry about him."
Also lurking was Brooks Koepka, who won a second straight US Open title last year then held off Woods to win the PGA Championship.
Koepka was alone in fourth on 206 after a three-under 69.
Former US Open champion Webb Simpson had an eagle and seven birdies in his 64 for 207, level with England's Ian Poulter, who signed for a 68.